The Monge-ampere Equation and Its Applications

By Alessio Figalli. 2017

257 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/900ed654b3e9b17a8548c4410432d041118cdd73

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

On the continuity of center-outward distribution and quantile functions (2018, background): “We note that, although several fundamental results have been obtained in the last 25 years on the regularity of optimal transport maps (see [8, 9] for a survey), the proof of the above-mentioned property is rather delicate, due to the fact that the density of Ud is singular at the origin whenever d ”
Statistical Aspects of Wasserstein Distances (2018, background): “The regularity result was discovered by Caﬀarelli (1992); see Figalli (2017) for an accessible exposition.”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, background): “It follows that φ is a.e. differentiable, with ‖∇φ(x)‖ ≤ 1, so that (Corollary (A.27) in Figalli (2017)), denoting by ∂φ(x) the subdifferential of φ at x,12 ∂φ(Rd) := ⋃ x∈Rd ∂φ(x) ⊆ Sd. (2.4) Moreover, since P has a density, Proposition 10 in McCann (1995) implies that 10Namely, the product of the…”
Global regularity for the Monge-Ampère equation with natural boundary condition (2018, methodology): “2 is based on delicate analysis on sub-level sets of the solution near the boundary and uses various techniques on the Monge-Ampère equation [16, 18], in particular those from Caffarelli’s papers.”
Plugin estimation of smooth optimal transport maps (2021, methodology): “Further smoothness properties of Brenier potentials, and therefore of optimal transport maps, have been studied via regularity theory of partial differential equations of the Monge-Ampère type, and we refer to De Philippis and Figalli (2014); Figalli (2017) for surveys.”
Stein kernels and moment maps (2018, methodology): “For background on regularity theory for Monge-Ampère PDEs, we refer to the lecture notes [16].”